I’ve spent a lot of time in math classrooms, and “4 is divisible by 2” is always taken to be equivalent to “4 divided by 2 -]has/-] [is] a quotient that is an integer.”
Colloquially, this is how people speak. But they are different sentences that mean different things. Again, the first is a predicative statement, the second is an identity statement using an operative desciption
If that is not what you mean, then the only other possibility is that you are taking “4 is divisible by 2” to mean “4 can be divided by 2” and asserting this as an intrinsic property of 4. f this is what you mean, this is what you you should say, because divisibility has a more specific meaning. Anyway, is that what you mean??
Yes.
this is a particularly lousy candidate for an intrinsic property of 4 since literally any number can be divided by 2.
Why is it lousy? Any object can share an intrinsic property with another. For example, we are both human. But that doesn’t mean “being human” is a lousy candidate for being an intrinsic property for both of us. And being human is certainly not a relation.
No, I am not saying that for 4 to have an essence it must stand in relation to other numbers. Not at all. I am saying that 4 has no essence to speak of–that all there is to know about 4 is how it relates to other things.
Now you are using an epistemological thesis for support of a metaphysical claim. Moreoever, the epistemological claim is false. I wouldn’t know the relations 5>4 is true and 4<5 is false without knowing what 4 and 5 means. Therefore, knowing what 4 and 5 means is epistemically prior to knowing the relations they stand in.
Yes, and I’ve addressed this before as well. The “why?” you are looking for is simply that 5>4 and 4>5 both have inferential relationships with other relations. We could only say 4>5 if we were willing to give up our beliefs about these other relations such as 4=2+2 and 4=3+2 and 3>2, but neither of us are willing to do so./
A relational statement is certainly
derivable from other relations, but if the truth-conditions for
all relations are in principle only dependent on other relations, you are left with certain assumptions within that system of inferences that cannot be accounted for by other relations. Spending enough time in mathematics, you should know this is precisely one of the consequences Godel demonstrated with his Incompleteness Theorem. Wikipedia, for instance, says
“the first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an “effective procedure” is capable of proving all facts about the natural numbers. For any such system, there
will always be statements about the natural numbers that are true, but that are **unprovable **within the system.”
Notice the bold-faced I highlighted. I take that as direct evidence that there are
brute facts about natural numbers the whole set of relations within that system cannot demonstrate. So your statement above is
demonstrably false.